P. M. Pankade 1, D. H. Tupe 2, G. R. Gandhe 3
|
|
- Rosa Rich
- 5 years ago
- Views:
Transcription
1 ISSN: Volume 5, Issue 5, May 6 Static Fleural Analysis of Thick Beam Using Hyperbolic Shear Deformation Theory P. M. Pankade, D. H. Tupe, G. R. Gandhe P.G. Student, Dept. of Civil Engineering, Deogiri Institute of engineering and Management Studies, Maharashtra, India. Assistant Professor, Dept. of Civil Engineering, Deogiri Institute of engineering and Management Studies, Maharashtra, India. Assistant Professor, Dept. of Civil Engineering, Deogiri Institute of engineering and Management Studies, Maharashtra, India. Abstract - In the present study, a hyperbolic shear deformation theory is developed for static fleural analysis of thick isotropic beams. Simply supported thick isotropic beams analyzed for the aial displacement, Transverse displacement, Aial bending stress and transverse shear stress. In this theory the hyperbolic sine and cosine function is used in the displacement field to represent the shear deformation effect and satisfy the zero transverse shear stress condition at top and bottom surface of the beams. The Governing differential equation and boundary conditions of the theory are obtained by using Principle of virtual work. The numerical results have been computed for various lengths to thickness ratios of the beams and the results obtained are compared with those of Elementary, Timoshenko, trigonometric and other higher order refined theories and with the available solution in the literature. Key Words - Isotropic beam, hyperbolic shear deformation theory, principle of virtual work, shear deformation, static fleure, transverse shear stress, thick beam. I. INTRODUCTION The wide spread use of shear fleible materials in aircraft, automotive, shipbuilding and other industries has stimulated interest in the accurate prediction of structural behaviour of beams. Theories of beams involve the reduction of a three dimensional problems of elasticity theory to a onedimensional problems. Since the thickness dimension is much smaller than the longitudinal dimension, it is possible to approimate the distribution of the displacement, strain and stress components in the thickness dimension. The various methods of development of refined theories based on the reduction of the three dimensional problems of mechanics of elastic bodies are discussed by Gol denveizer [], Kil chevskiy [], Donnell [], Vlasov and Leontev [4], Sayir and Mitropoulos [5]. It is well-known that elementary theory of bending of beam based on Euler-Bernoulli hypothesis that the plane sections which are perpendicular to the neutral layer before bending remain plane and perpendicular to the neutral layer after bending, implying that the transverse shear and transverse normal strains are zero. Thus, the theory disregards the effects of the shear deformation. It is also known as classical beam theory. The theory is applicable to slender beams and should not be applied to thick or deep beams. When elementary theory of beam () is used for the analysis thick beams, deflections are underestimated and natural frequencies and buckling loads are overestimated. This is the consequence of neglecting transverse shear deformations in. Rankine [6], Bresse [7] were the first to include both the rotatory inertia and shear fleibility effects as refined dynamical effects in beam theory. This theory is, referred as Timoshenko beam theory as mentioned in the literature by Rebello, et.al. [8] and based upon kinematics it is known as first-order shear deformation theory (). Stephen and Levinson [9] have introduced a refined theory incorporating shear curvature, transverse direct stress and rotatory inertia effects. The limitations of the elementary theory of bending () of beams and first order shear deformation theory () for beams forced the development of higher order shear deformation theories. Ghugal and Dahake [] have developed a trigonometric shear deformation theory for fleure of thick or deep beams, taking into account transverse shear deformation effect. The number of variables in the present theory is same as that in the first order shear deformation theory. The sinusoidal function is used in displacement field in terms of thickness coordinate to represent the shear deformation effects. This theory obviates the need of shear correction factor. Ghugal and Sharma, Sayyad and Ghugal developed a variationally consistent refined hyperbolic shear deformation theory for fleure and free vibration of thick isotropic beam. This theory takes into account transverse shear deformations effects. In this paper, a hyperbolic shear deformation theory is developed for static fleural analysis of thick isotropic beams. The theory is applied to a Simply supported thick isotropic beams to analysed the aial displacement, Transverse displacement, Aial bending stress and transverse shear stress. The numerical results obtained for various lengths to thickness ratios of the beams and the results obtained are compared with those of Elementary, Timoshenko, Trigonometric and other higher order refined theories and with the available solution in the literature. II. FORMULATION OF PROBLEM Consider a thick isotropic Cantilever beam of length L in direction, Width b in y direction and depth h as shown in Figure-. Where, y, z are Cartesian coordinates. The beam is subjected to transverse load of intensity q() per unit length of beam. Under this condition, the aial displacement, Transverse displacement, Aial bending stress and transverse shear stress are required to be determined. A. Assumptions made in the theoretical formulation:. The aial displacement (u) consist of two parts: All Rights Reserved 6 IJSETR 46
2 a. Displacement given by elementary theory of bending. b. Displacement due to shear deformation, which is assume to be hyperbolic in nature with respect to thickness coordinate.. The transverse displacement (w) in z direction is assumed to be function of coordinate.. One-dimensional constitutive laws are used. 4. The beam is subjected to lateral load only. ISSN: Volume 5, Issue 5, May 6 Shear strain: u w z z z z z cosh 4 cosh h h [4] Stresses: The one-dimensional Hooke s law is applied for isotropic material, stress is related to strain and shear stress is related to shear strain by the following constitutive relations. E u 4 z z ze E hsinh cosh h h z Gz z z G cosh 4 cosh h h [5] [6] Where E and G are the elastic constants of the beam material. Fig-: Cantilever beam bending under -z plane B. The Displacement Field: Based on the above mentioned assumptions, the displacement field of the present beam theory can be epressed as follows. The hyperbolic function is assigned according to the shearing stress distribution through the thickness of beam. w u, z, t z, t z 4 z hsinh cosh, t h h w(, t) w(, t) Where, u = Aial displacement in direction which is a function of, z and t. w = Transverse displacement in z direction which is function of and t. = Rotation of cross section of beam at neutral ais due to shear which is an unknown function to be determined and it is function of and t. Normal strain: u u z 4 z z hsinh cosh h h [] [] [] C. Governing Differential Equations: Governing differential equations and boundary conditions are obtained from Principle of virtual work. Using equations for stresses, strains and principle of virtual work. Using equations for stresses, strains and principle of virtual work, variationally consistent differential equations for beam under consideration are obtained. The principle of virtual work when applied to beam leads to: b L zh/ zh/ L zh/ u w b. u. w ddz t t zh/ L. z. z ddz q wd Where δ = variational operator. Employing Greens theorem in above equation successively, we obtained the coupled Euler-Langrange equations, which are the governing differential equations and associated boundary conditions of the beam. The governing differential equations obtained are as follows: 4 4 w w 4 EI A I A t t t w A q, t t w w EI A B I A B t t t GAC [7] [8] [9] All Rights Reserved 6 IJSETR 464
3 Where A, B and C are the stiffness coeffcients in governing equations. The associated consistent natural boundary conditions obtained are of following form along the edges = and = L. d w d d w d EI A I A d d ddt dt d w d EI A d d EI A B d d d w d [] Where w is prescribed. [] Where dw is prescribed. d [] Where is prescribed. The fleural behaviour of beam is given by solution of above equations 8 and 9 by discarding all terms containing time derivatives and satisfying the associate boundary conditions. The stiffness coefficient used in governing equations 8, 9,, and are described as below: A cosh 4sinh cosh 5 6 sinh cosh B 4sinh cosh cosh sinh 6 cosh C 6sinh cosh cosh 5 [] [4] [5] ISSN: Volume 5, Issue 5, May 6 The general solution of equation 9 is as follows: Q kcosh ksinh [] D Where the constants α, β, λ and D used in above equations are given below: B GAC A, A DA, D EI The equation of transverse displacement w() is obtained by substituting the epression of φ() in equation 8 and integrating it thrice with respect to. The general solution for w() is obtained as follows: EIw D B qdddd A k k sinh k cosh k4 k5 k [] Where k, k, k, k 4, k 5 and k 6 are the constants of integration and can be obtained by applying the boundary conditions of the beams.. ILLUSTARTIVE EXAMPLE In order to prove the efficiency of the present theory, the following numerical eamples are considered. The following material properties for beam are used. Material properties:. Modulus of Elasticity E = GPa. Poissions ratio =.. Density =78 Kg/m D. The General solution of Governing equilibrium equations of beam: The general solution for transverse displacement w() and φ() can be obtained from equation 8 and 9 by discarding the terms containing time (t) derivatives. Integrating and rearranging the equation 8, we obtained the following equation d w d Q A d d D Where, Q() is generalized shear force for beam. [6] A. Eample : Cantilever beam with uniformly load q() = q () A cantilever beam with the origin of beam on left end support at =. The beam is subjected to uniformly distributed load of q() over the span L on surface z = h/ acting in the z direction is given by, The boundary conditions associated with this Problem are as follows: At fied end ( = L): q q ( ) Q qd k The second governing equation 9 can be written as: [7] dw EI d dw EI d d EI d d EI d d w B d [8] d A d Now using equations 6 and 8 a single equation in terms of φ is obtained as: d d Q [9] D At free end ( = ): EI EIw All Rights Reserved 6 IJSETR dw EI d 465
4 General epressions obtained for w() and φ() are as follows: A q L sinh cosh C Gbh L 5 B E h w 5 L L L C G L L 6 L A 5 E h cosh sinh G L C L L The aial displacement, stresses and transverse shear stress obtained based on above solutions are as follows: 4 E h B z L L L L G L C L L u h h Eh A 5 sinh cosh G L C A E L z 4 z sinh cosh sinh cosh C G h h h L E h B 6 4 z L L L G L C L h h Eh A 5 Lcosh Lsinh G L C A E z 4 z sinh cosh L cosh L sinh C G h h L z A Eh C G L z 6 6 z L G L C 4 Eh G L C L sinh L cosh z cosh 6 cosh cosh h 4 z 4 48 h A L z z cosh 4 cosh C h h h sinh cosh L IV. NUMERICAL RESULTS The numerical results for aial displacement, transverse displacement, bending stress and transverse shear stress are presented in following non-dimensional form and the values are presented in Table- and Table - E h L 8 h h A 5 L sinh L cosh B [] [] [4] [5] [6] [7] ISSN: Volume 5, Issue 5, May 6 Table-I: Non-Dimensional Aial Displacement u at (=, z=h/), Transverse Deflection w at (=L, z=), Aial Stress σ at (=, z=h/), Maimum Transverse Shear Stresses τ z and τ z (=, z=) of the cantilever Beam Subjected to Varying Load for Aspect Ratio 4. Source Model w u σ τz Present HPSDT Dahake Krishna Murty Timoshenko Bernoulli-Euler Table-II: Non-Dimensional Aial Displacement u at (=, z=h/), Transverse Deflection w at (=L, z=), Aial Stress σ at (=, z=h/), Maimum Transverse Shear Stresses τ z and τ z (=, z=) of the cantilever Beam Subjected to Varying Load for Aspect Ratio. Source Model w u σ τz Present HPSDT Dahake Krishna Murty Timoshenko Bernoulli-Euler Fig-: Variation of Transverse Displacement w τz τz AS Ebh Eb w ; u u q L q h 4 b bz ; z q q All Rights Reserved 6 IJSETR 466
5 ISSN: Volume 5, Issue 5, May Present-HPSDT Fig-: Variation of Maimum Aial displacement u for AS Fig-5: Variation of Maimum Aial stress σ for AS Fig-4: Variation of Maimum Aial displacement u for AS Fig-6: Variation of Maimum Aial stress σ for AS All Rights Reserved 6 IJSETR 467
6 ISSN: Volume 5, Issue 5, May Fig-7: Variation of Transverse shear stress τ z for AS Fig-9: Variation of Transverse shear stress τ z for AS Fig-8: Variation of Transverse shear stress τ z for AS V. CONCLUDING REMARK From the static fleural analysis of cantilever beam following conclusion are drawn:. The result of maimum transverse displacement w obtained by present theory is in ecellent agreement with those of other equivalent refined and higher order theories. The variation of for AS 4 and are presented as shown in Fig-.. From Fig- and Fig-4, it can be observed that, the result of aial displacement u for beam subjected to uniform load varies linearly through the thickness of beam for AS 4 and respectively. -.8 Fig-: Variation of Transverse shear stress τ z for AS. The maimum Non-dimensional aial stresses σ for AS 4 and varies linearly through the thickness of beam as shown in Figure 5 and Figure 6 respectively. 4. The transverse shear stresses τ z and τ z are obtained directly by constitutive relation. Fig-7, 8, 9 and Fig- shows the through thickness variation of transverse shear stress for thick isotropic beam for AS 4 and. From this fig it can be observed that, the transverse shear stress satisfy the zero condition at top (z=h/) and at bottom (z=-h/) surface of the beam. All Rights Reserved 6 IJSETR 468
7 REFERENCES [] Goldenviezer, A. L., Methods for Justifying and Refining the Theory Shells, Journal of Applied Mathematics and Mechanics, Vol., No.4, pp , 968. [] Kilchevskiy, N. A., Fundamentals of the Analytical Mechanics of Shells, NASA TT F-9, Washington, D.C., pp. 8-7, 965. [] Donnell L. H., Beams, Plates and Shells, McGraw-Hill Book Company. New York, 976. [4] Vlasov V. Z., and Leontev, U. N., Beams, Plates and Shells on Elastic Foundations, Translated from Russian by Barouch, A., and edited by Pelz, T. Israel program for scientific translations Ltd., Jerusalem, Chapter, pp. -8, 96. [5] Sayir M., and Mitropoulos, C., On Elementary Theories of Linear Elastic Beams, Plates and Shells, Zeitschrift fur Angewandte Mathematic und Physic, Vol., No., pp. -55, 98. [6] Rankine W. J. M., A Manual of Applied Mechanics, R. Griffin and Company Ltd., London, U. K., pp. 4-44, 858. [7] Bresse J. A. C., Cours de Mecanique Applique, Paris: Mallet-bachelier, (866 nd ed.), Gauthier-Villars, Paris, 859. [8] Rebello C. A., Bert, C. W., and Gordaninejad, F., Vibration of Bi modular Sandwich Beams with Thick Facings: A New Theory and Eperimental Results, Journal of Sound and Vibration, Vol. 9, No., pp. 8-97, 98. [9] Stephen, N. G., and Levinson, M., A Second Order Beam Theory, Journal of Sound and Vibration, Vol. 67, pp. 9-5, 979. [] Ajay G. Dahake, Dr. Yuwaraj M. Ghugal Fleure of Thick Simply Supported Beam Using Trigonometric Shear Deformation Theory, International Journal of Scientific and Research Publications, Volume, Issue, November. [] Levinson M., A New Rectangular Beam Theory, Journal of Sound and Vibration, Vol. 74, pp. 8-87, 98. [] Tupe D.H. and Dahake A.G Trigonometric Shear Deformation Theory For Thick Simply Supported Beams, International Journal of Re- search in Engineering and Technology, Vol. 4, No., pp , 5. ISSN: Volume 5, Issue 5, May 6 [] Levinson M., Further Results of a New Beam Theory, Journal of Sound and Vibration, Vol. 77, pp , 98. [4] Levinson M., On Bickfords Consistent Higher Order Beam Theory, Mechanics Research Communications, Vol., pp.-9, 985. [5] Krishna Murty, A. V., Toward a Consistent Beam Theory, AIAA Journal, Vol., pp. 8-86, 984. [6] Ajay G. Dahake, Dr. Yuwaraj M. Ghugal Fleure of Thick Simply Supported Beam Using Trigonometric Shear Deformation Theory, International Journal of Scientific and Research Publications, Volume, Issue, November. [7] Ajay G. Dahake, Dr. Yuwaraj M. Ghugal Fleure of thick beams using refined shear deformation theory, International Journal of civil and structural engineering, Volume, No.,. [8] Yuwaraj M. Ghugal, Ajay G. Dahake Fleure of Simply Supported Thick Beams Using Refined Shear Deformation Theory, World Academy of Science, Engineering and Technology International Journal of Civil, Structural, Construction and Architectural Engineering Vol:7, No:,. [9] Ghugal Y. M. and Sharma, R., A Hyperbolic Shear Deformation Theory for Fleure and Vibration of Thick Isotropic Beams, International Journal of Computational Methods, Vol. 6, No. 4, pp , 9. [] Ghugal, Y. M. and Sharma, R., A Refined Shear Deformation Theory for Fleure of Thick Beams, Latin American Journal of Solids and Structures, Vol. 8, pp. 8-9,. [] Sayyad A. S. and Ghugal Y. M., Fleure of Thick Beams using New Hyperbolic Shear De- formation Theory, International Journal of Mechanics, Vol. 5, No., pp. -,. [] Nimbalkar V. N. and Dahake A.G Displacement and stresses for Thick Beam using New Hyperbolic Shear Deformation Theory, International Journal of pure and applied research in Engineering and technology, Vol., No. 9, pp. -, 5. All Rights Reserved 6 IJSETR 469
Pune, Maharashtra, India
Volume 6, Issue 6, May 17, ISSN: 78 7798 STATIC FLEXURAL ANALYSIS OF THICK BEAM BY HYPERBOLIC SHEAR DEFORMATION THEORY Darakh P. G. 1, Dr. Bajad M. N. 1 P.G. Student, Dept. Of Civil Engineering, Sinhgad
More informationFlexure of Thick Cantilever Beam using Third Order Shear Deformation Theory
International Journal of Engineering Research and Development e-issn: 78-67X, p-issn: 78-8X, www.ijerd.com Volume 6, Issue 1 (April 13), PP. 9-14 Fleure of Thick Cantilever Beam using Third Order Shear
More informationAnalysis of Thick Cantilever Beam Using New Hyperbolic Shear Deformation Theory
International Journal of Research in Advent Technology, Vol.4, No.5, May 16 E-ISSN: 1-967 Analysis of Thick Cantilever Beam Using New Hyperbolic Shear Deformation Theory Mr. Mithun. K. Sawant 1, Dr. Ajay.
More informationFlexural Analysis of Deep Aluminum Beam
Journal of Soft Computing in Civil Engineering -1 (018) 71-84 journal homepage: http://www.jsoftcivil.com/ Fleural Analysis of Deep Aluminum Beam P. Kapdis 1, U. Kalwane 1, U. Salunkhe 1 and A. Dahake
More informationFlexural analysis of deep beam subjected to parabolic load using refined shear deformation theory
Applied and Computational Mechanics 6 (2012) 163 172 Flexural analysis of deep beam subjected to parabolic load using refined shear deformation theory Y. M. Ghugal a,,a.g.dahake b a Applied Mechanics Department,
More informationMODIFIED HYPERBOLIC SHEAR DEFORMATION THEORY FOR STATIC FLEXURE ANALYSIS OF THICK ISOTROPIC BEAM
MODIFIED HYPERBOLIC SHEAR DEFORMATION THEORY FOR STATIC FLEXURE ANALYSIS OF THICK ISOTROPIC BEAM S. Jasotharan * and I.R.A. Weerasekera University of Moratuwa, Moratuwa, Sri Lanka * E-Mail: jasos91@hotmail.com,
More informationLecture 15 Strain and stress in beams
Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME
More informationFlexure of Thick Simply Supported Beam Using Trigonometric Shear Deformation Theory
International Jornal of Scientific and Research Pblications, Volme, Isse 11, November 1 1 ISSN 5-15 Flere of Thick Simply Spported Beam Using Trigonometric Shear Deformation Theory Ajay G. Dahake *, Dr.
More informationBending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theory
J. Appl. Comput. ech., 4(3) (018) 187-01 DOI: 10.055/JAC.017.3057.1148 ISSN: 383-4536 jacm.scu.ac.ir Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theor
More information7.4 The Elementary Beam Theory
7.4 The Elementary Beam Theory In this section, problems involving long and slender beams are addressed. s with pressure vessels, the geometry of the beam, and the specific type of loading which will be
More information4.5 The framework element stiffness matrix
45 The framework element stiffness matri Consider a 1 degree-of-freedom element that is straight prismatic and symmetric about both principal cross-sectional aes For such a section the shear center coincides
More informationINTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2011
INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volue, No 1, 11 Copyright 1 All rights reserved Integrated Publishing services Research article ISSN 976 499 Coparison of various shear deforation
More informationPresented By: EAS 6939 Aerospace Structural Composites
A Beam Theory for Laminated Composites and Application to Torsion Problems Dr. BhavaniV. Sankar Presented By: Sameer Luthra EAS 6939 Aerospace Structural Composites 1 Introduction Composite beams have
More informationComb Resonator Design (2)
Lecture 6: Comb Resonator Design () -Intro. to Mechanics of Materials Sh School of felectrical ti lengineering i and dcomputer Science, Si Seoul National University Nano/Micro Systems & Controls Laboratory
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationLongitudinal buckling of slender pressurised tubes
Fluid Structure Interaction VII 133 Longitudinal buckling of slender pressurised tubes S. Syngellakis Wesse Institute of Technology, UK Abstract This paper is concerned with Euler buckling of long slender
More informationEffect of Specimen Dimensions on Flexural Modulus in a 3-Point Bending Test
Effect of Specimen Dimensions on Flexural Modulus in a 3-Point Bending Test M. Praveen Kumar 1 and V. Balakrishna Murthy 2* 1 Mechanical Engineering Department, P.V.P. Siddhartha Institute of Technology,
More informationA NEW REFINED THEORY OF PLATES WITH TRANSVERSE SHEAR DEFORMATION FOR MODERATELY THICK AND THICK PLATES
A NEW REFINED THEORY OF PLATES WITH TRANSVERSE SHEAR DEFORMATION FOR MODERATELY THICK AND THICK PLATES J.M. MARTÍNEZ VALLE Mechanics Department, EPS; Leonardo da Vinci Building, Rabanales Campus, Cordoba
More informationMacaulay s method for a Timoshenko beam
Macaulay s method for a Timoshenko beam N. G. Stephen School of Engineering Sciences, Mechanical Engineering, University of Southampton, Highfield, Southampton SO7 BJ, UK E-mail: ngs@soton.ac.uk Abstract
More informationTheories of Straight Beams
EVPM3ed02 2016/6/10 7:20 page 71 #25 This is a part of the revised chapter in the new edition of the tetbook Energy Principles and Variational Methods in pplied Mechanics, which will appear in 2017. These
More informationAircraft Structures Structural & Loading Discontinuities
Universit of Liège Aerospace & Mechanical Engineering Aircraft Structures Structural & Loading Discontinuities Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/
More informationFINITE ELEMENT ANALYSIS OF BEAM
YMCA University of Science & Technology, Faridabad, Haryana, Oct 9-, FINITE ANALYSIS OF Hasan Zakir Jafri, I.A. Khan, S.M. Muzakkir, Research Scholar, Faculty of Engineering & Technology, Jamia Millia
More informationIntroduction to Finite Element Method. Dr. Aamer Haque
Introduction to Finite Element Method 4 th Order Beam Equation Dr. Aamer Haque http://math.iit.edu/~ahaque6 ahaque7@iit.edu Illinois Institute of Technology July 1, 009 Outline Euler-Bernoulli Beams Assumptions
More informationChapter 5 Structural Elements: The truss & beam elements
Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations
More informationStress Analysis of Laminated Composite and Sandwich Beams using a Novel Shear and Normal Deformation Theory
134 Stress Analysis of Laminated Composite and Sandwich Beams using a Novel Shear and Normal Deformation Theory Abstract A novel Normal and Shear Deformation Theory (NSDT) for analysis of laminated composite
More informationStructures. Carol Livermore Massachusetts Institute of Technology
C. Livermore: 6.777J/.7J Spring 007, Lecture 7 - Structures Carol Livermore Massachusetts Institute of Technolog * With thanks to Steve Senturia, from whose lecture notes some of these materials are adapted.
More informationAnalytical Strip Method for Thin Isotropic Cylindrical Shells
IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) e-issn: 2278-1684,p-ISSN: 2320-334X, Volume 14, Issue 4 Ver. III (Jul. Aug. 2017), PP 24-38 www.iosrjournals.org Analytical Strip Method for
More informationModule 2 Stresses in machine elements
Module 2 Stresses in machine elements Lesson 3 Strain analysis Instructional Objectives At the end of this lesson, the student should learn Normal and shear strains. 3-D strain matri. Constitutive equation;
More informationStatic Analysis of Cylindrical Shells
Static Analysis of Cylindrical Shells Akshaya Dhananjay Patil 1, Mayuri Madhukar Jadhav 2 1,2 Assistant Professor, Dept. of Civil Engineering, Padmabhooshan Vasantraodada Patil Institute Of Technology,
More informationBending and free vibration analysis of thick isotropic plates by using exponential shear deformation theory
Applied and Computational Mechanics 6 (01 65 8 Bending and free vibration analysis of thick isotropic plates by using exponential shear deformation theory A. S. Sayyad a,, Y. M. Ghugal b a Department of
More informationSection 6: PRISMATIC BEAMS. Beam Theory
Beam Theory There are two types of beam theory aailable to craft beam element formulations from. They are Bernoulli-Euler beam theory Timoshenko beam theory One learns the details of Bernoulli-Euler beam
More informationModeling of the Bending Stiffness of a Bimaterial Beam by the Approximation of One-Dimensional of Laminated Theory
. Flores-Domínguez Int. Journal of Engineering Research and Applications RESEARCH ARTICLE OPEN ACCESS odeling of the Bending Stiffness of a Bimaterial Beam by the Approimation of One-Dimensional of Laminated
More informationBending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory
Curved and Layer. Struct. 215; 2:279 289 Reseach Article Open Access A. S. Sayyad*, Y. M. Ghugal, and N. S. Naik Bending analysis of laminated composite and sandwich beams according to refined trigonometric
More informationSERVICEABILITY OF BEAMS AND ONE-WAY SLABS
CHAPTER REINFORCED CONCRETE Reinforced Concrete Design A Fundamental Approach - Fifth Edition Fifth Edition SERVICEABILITY OF BEAMS AND ONE-WAY SLABS A. J. Clark School of Engineering Department of Civil
More informationMECHANICS OF MATERIALS Sample Problem 4.2
Sample Problem 4. SOLUTON: Based on the cross section geometry, calculate the location of the section centroid and moment of inertia. ya ( + Y Ad ) A A cast-iron machine part is acted upon by a kn-m couple.
More informationRigid and Braced Frames
RH 331 Note Set 12.1 F2014abn Rigid and raced Frames Notation: E = modulus of elasticit or Young s modulus F = force component in the direction F = force component in the direction FD = free bod diagram
More informationStructural Analysis. For. Civil Engineering.
Structural Analysis For Civil Engineering By www.thegateacademy.com ` Syllabus for Structural Analysis Syllabus Statically Determinate and Indeterminate Structures by Force/ Energy Methods; Method of Superposition;
More informationComb resonator design (2)
Lecture 6: Comb resonator design () -Intro Intro. to Mechanics of Materials School of Electrical l Engineering i and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory
More informationAPPLICATION OF DIRECT VARIATIONAL METHOD IN THE ANALYSIS OF ISOTROPIC THIN RECTANGULAR PLATES
VOL. 7, NO. 9, SEPTEMBER 0 ISSN 89-6608 006-0 Asian Research Publishing Network (ARPN). All rights reserved. APPLICATION OF DIRECT VARIATIONAL METHOD IN THE ANALYSIS OF ISOTROPIC THIN RECTANGULAR PLATES
More informationCHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES
CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES * Governing equations in beam and plate bending ** Solution by superposition 1.1 From Beam Bending to Plate Bending 1.2 Governing Equations For Symmetric
More informationStability of Simply Supported Square Plate with Concentric Cutout
International OPEN ACCESS Journal Of Modern Engineering Research (IJMER) Stability of Simply Supported Square Plate with Concentric Cutout Jayashankarbabu B. S. 1, Dr. Karisiddappa 1 (Civil Engineering
More information7. Hierarchical modeling examples
7. Hierarchical modeling examples The objective of this chapter is to apply the hierarchical modeling approach discussed in Chapter 1 to three selected problems using the mathematical models studied in
More informationCHAPTER 5. Beam Theory
CHPTER 5. Beam Theory SangJoon Shin School of Mechanical and erospace Engineering Seoul National University ctive eroelasticity and Rotorcraft Lab. 5. The Euler-Bernoulli assumptions One of its dimensions
More informationCOURSE TITLE : APPLIED MECHANICS & STRENGTH OF MATERIALS COURSE CODE : 4017 COURSE CATEGORY : A PERIODS/WEEK : 6 PERIODS/ SEMESTER : 108 CREDITS : 5
COURSE TITLE : APPLIED MECHANICS & STRENGTH OF MATERIALS COURSE CODE : 4017 COURSE CATEGORY : A PERIODS/WEEK : 6 PERIODS/ SEMESTER : 108 CREDITS : 5 TIME SCHEDULE MODULE TOPICS PERIODS 1 Simple stresses
More informationCOPYRIGHTED MATERIAL. Index
Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,
More informationIraq Ref. & Air. Cond. Dept/ Technical College / Kirkuk
International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-015 1678 Study the Increasing of the Cantilever Plate Stiffness by Using s Jawdat Ali Yakoob Iesam Jondi Hasan Ass.
More informationMechanical Design in Optical Engineering
OPTI Buckling Buckling and Stability: As we learned in the previous lectures, structures may fail in a variety of ways, depending on the materials, load and support conditions. We had two primary concerns:
More informationCOMPARATIVE STUDY OF VARIOUS BEAMS UNDER DIFFERENT LOADING CONDITION USING FINITE ELEMENT METHOD
COMPARATIVE STUDY OF VARIOUS BEAMS UNDER DIFFERENT LOADING CONDITION USING FINITE ELEMENT METHOD A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Bachelor of Technology
More informationBending of Simply Supported Isotropic and Composite Laminate Plates
Bending of Simply Supported Isotropic and Composite Laminate Plates Ernesto Gutierrez-Miravete 1 Isotropic Plates Consider simply a supported rectangular plate of isotropic material (length a, width b,
More informationNumerical and experimental analysis of a cantilever beam: A laboratory project to introduce geometric nonlinearity in Mechanics of Materials
Numerical and experimental analysis of a cantilever beam: A laboratory project to introduce geometric nonlinearity in Mechanics of Materials Tarsicio Beléndez (1) and Augusto Beléndez (2) (1) Departamento
More informationCHAPTER -6- BENDING Part -1-
Ishik University / Sulaimani Civil Engineering Department Mechanics of Materials CE 211 CHAPTER -6- BENDING Part -1-1 CHAPTER -6- Bending Outlines of this chapter: 6.1. Chapter Objectives 6.2. Shear and
More informationNote on Mathematical Development of Plate Theories
Advanced Studies in Theoretical Phsics Vol. 9, 015, no. 1, 47-55 HIKARI Ltd,.m-hikari.com http://d.doi.org/10.1988/astp.015.411150 Note on athematical Development of Plate Theories Patiphan Chantaraichit
More information1 Static Plastic Behaviour of Beams
1 Static Plastic Behaviour of Beams 1.1 Introduction Many ductile materials which are used in engineering practice have a considerable reserve capacity beyond the initial yield condition. The uniaxial
More informationAdvanced Structural Analysis EGF Section Properties and Bending
Advanced Structural Analysis EGF316 3. Section Properties and Bending 3.1 Loads in beams When we analyse beams, we need to consider various types of loads acting on them, for example, axial forces, shear
More informationA two variable refined plate theory for orthotropic plate analysis
A two variable refined plate theory for orthotropic plate analysis R.P. Shimpi *, H.G. Patel Department of Aerospace Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, Maharashtra,
More informationProblem d d d B C E D. 0.8d. Additional lecturebook examples 29 ME 323
Problem 9.1 Two beam segments, AC and CD, are connected together at C by a frictionless pin. Segment CD is cantilevered from a rigid support at D, and segment AC has a roller support at A. a) Determine
More informationStress Functions. First Semester, Academic Year 2012 Department of Mechanical Engineering Chulalongkorn University
Lecture Note 3 Stress Functions First Semester, Academic Year 01 Department of Mechanical Engineering Chulalongkorn Universit 1 Objectives #1 Describe the equation of equilibrium using D/3D stress elements
More informationDeflection profile analysis of beams on two-parameter elastic subgrade
1(213) 263 282 Deflection profile analysis of beams on two-parameter elastic subgrade Abstract A procedure involving spectral Galerkin and integral transformation methods has been developed and applied
More informationMechanics of Inflatable Fabric Beams
Copyright c 2008 ICCES ICCES, vol.5, no.2, pp.93-98 Mechanics of Inflatable Fabric Beams C. Wielgosz 1,J.C.Thomas 1,A.LeVan 1 Summary In this paper we present a summary of the behaviour of inflatable fabric
More informationPURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC.
BENDING STRESS The effect of a bending moment applied to a cross-section of a beam is to induce a state of stress across that section. These stresses are known as bending stresses and they act normally
More information: APPLIED MECHANICS & STRENGTH OF MATERIALS COURSE CODE : 4021 COURSE CATEGORY : A PERIODS/ WEEK : 5 PERIODS/ SEMESTER : 75 CREDIT : 5 TIME SCHEDULE
COURSE TITLE : APPLIED MECHANICS & STRENGTH OF MATERIALS COURSE CODE : 4021 COURSE CATEGORY : A PERIODS/ WEEK : 5 PERIODS/ SEMESTER : 75 CREDIT : 5 TIME SCHEDULE MODULE TOPIC PERIODS 1 Simple stresses
More informationChapter 3. Load and Stress Analysis
Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3
More information202 Index. failure, 26 field equation, 122 force, 1
Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic
More informationP-I diagrams for linear-elastic cantilevered Timoshenko beams including higher modes of vibration
P-I diagrams for linear-elastic cantilevered Timoshenko beams including higher modes of vibration L. J. van der Meer, J. G. M. Kerstens and M. C. M. Bakker Faculty of Architecture, Building and Planning,
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 11 Last class, what we did is, we looked at a method called superposition
More informationChapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship
Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction
More information1859. Forced transverse vibration analysis of a Rayleigh double-beam system with a Pasternak middle layer subjected to compressive axial load
1859. Forced transverse vibration analysis of a Rayleigh double-beam system with a Pasternak middle layer subjected to compressive axial load Nader Mohammadi 1, Mehrdad Nasirshoaibi 2 Department of Mechanical
More informationNonlinear bending analysis of laminated composite stiffened plates
Nonlinear bending analysis of laminated composite stiffened plates * S.N.Patel 1) 1) Dept. of Civi Engineering, BITS Pilani, Pilani Campus, Pilani-333031, (Raj), India 1) shuvendu@pilani.bits-pilani.ac.in
More informationApplication of Finite Element Method to Create Animated Simulation of Beam Analysis for the Course of Mechanics of Materials
International Conference on Engineering Education and Research "Progress Through Partnership" 4 VSB-TUO, Ostrava, ISSN 156-35 Application of Finite Element Method to Create Animated Simulation of Beam
More informationEE C245 ME C218 Introduction to MEMS Design
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 16: Energy
More informationVIBRATION ANALYSIS OF EULER AND TIMOSHENKO BEAMS USING DIFFERENTIAL TRANSFORMATION METHOD
VIBRATION ANALYSIS OF EULER AND TIMOSHENKO BEAMS USING DIFFERENTIAL TRANSFORMATION METHOD Dona Varghese 1, M.G Rajendran 2 1 P G student, School of Civil Engineering, 2 Professor, School of Civil Engineering
More informationFree vibration analysis of beams by using a third-order shear deformation theory
Sādhanā Vol. 32, Part 3, June 2007, pp. 167 179. Printed in India Free vibration analysis of beams by using a third-order shear deformation theory MESUT ŞİMŞEK and TURGUT KOCTÜRK Department of Civil Engineering,
More informationLecture notes Models of Mechanics
Lecture notes Models of Mechanics Anders Klarbring Division of Mechanics, Linköping University, Sweden Lecture 7: Small deformation theories Klarbring (Mechanics, LiU) Lecture notes Linköping 2012 1 /
More informationSymmetric Bending of Beams
Symmetric Bending of Beams beam is any long structural member on which loads act perpendicular to the longitudinal axis. Learning objectives Understand the theory, its limitations and its applications
More information3. BEAMS: STRAIN, STRESS, DEFLECTIONS
3. BEAMS: STRAIN, STRESS, DEFLECTIONS The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets
More informationPLAT DAN CANGKANG (TKS 4219)
PLAT DAN CANGKANG (TKS 4219) SESI I: PLATES Dr.Eng. Achfas Zacoeb Dept. of Civil Engineering Brawijaya University INTRODUCTION Plates are straight, plane, two-dimensional structural components of which
More informationBending Analysis of Symmetrically Laminated Plates
Leonardo Journal of Sciences ISSN 1583-0233 Issue 16, January-June 2010 p. 105-116 Bending Analysis of Symmetrically Laminated Plates Bouazza MOKHTAR 1, Hammadi FODIL 2 and Khadir MOSTAPHA 2 1 Department
More informationAircraft Structures Kirchhoff-Love Plates
University of Liège erospace & Mechanical Engineering ircraft Structures Kirchhoff-Love Plates Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/ Chemin
More informationThis version was downloaded from Northumbria Research Link:
Citation: Thai, Huu-Tai and Vo, Thuc (01) Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. International Journal of Mechanical Sciences,
More informationNonlinear RC beam element model under combined action of axial, bending and shear
icccbe 21 Nottingham University Press Proceedings of the International Conference on Computing in Civil and Building ngineering W Tiani (ditor) Nonlinear RC beam element model under combined action of
More information2. Determine the deflection at C of the beam given in fig below. Use principal of virtual work. W L/2 B A L C
CE-1259, Strength of Materials UNIT I STRESS, STRAIN DEFORMATION OF SOLIDS Part -A 1. Define strain energy density. 2. State Maxwell s reciprocal theorem. 3. Define proof resilience. 4. State Castigliano
More informationENGINEERING MECHANICS
ENGINEERING MECHANICS Engineering Mechanics Volume 2: Stresses, Strains, Displacements by C. HARTSUIJKER Delft University of Technology, Delft, The Netherlands and J.W. WELLEMAN Delft University of Technology,
More informationConsider an elastic spring as shown in the Fig.2.4. When the spring is slowly
.3 Strain Energy Consider an elastic spring as shown in the Fig..4. When the spring is slowly pulled, it deflects by a small amount u 1. When the load is removed from the spring, it goes back to the original
More informationDETAILED SYLLABUS FOR DISTANCE EDUCATION. Diploma. (Three Years Semester Scheme) Diploma in Architecture (DARC)
DETAILED SYLLABUS FOR DISTANCE EDUCATION Diploma (Three Years Semester Scheme) Diploma in Architecture (DARC) COURSE TITLE DURATION : Diploma in ARCHITECTURE (DARC) : 03 Years (Semester System) FOURTH
More informationA *69>H>N6 #DJGC6A DG C<>C::G>C<,8>:C8:H /DA 'D 2:6G - ( - ) +"' ( + -"( (' (& -+" % '('%"' +"-2 ( -!"',- % )% -.C>K:GH>IN D; AF69>HH>6,-+
The primary objective is to determine whether the structural efficiency of plates can be improved with variable thickness The large displacement analysis of steel plate with variable thickness at direction
More informationPES Institute of Technology
PES Institute of Technology Bangalore south campus, Bangalore-5460100 Department of Mechanical Engineering Faculty name : Madhu M Date: 29/06/2012 SEM : 3 rd A SEC Subject : MECHANICS OF MATERIALS Subject
More informationAPPLICATION OF THE GALERKIN-VLASOV METHOD TO THE FLEXURAL ANALYSIS OF SIMPLY SUPPORTED RECTANGULAR KIRCHHOFF PLATES UNDER UNIFORM LOADS
Nigerian Journal of Technology (NIJOTECH) Vol. 35, No. 4, October 2016, pp. 732 738 Copyright Faculty of Engineering, University of Nigeria, Nsukka, Print ISSN: 0331-8443, Electronic ISSN: 2467-8821 www.nijotech.com
More informationREVIEW FOR EXAM II. Dr. Ibrahim A. Assakkaf SPRING 2002
REVIEW FOR EXM II. J. Clark School of Engineering Department of Civil and Environmental Engineering b Dr. Ibrahim. ssakkaf SPRING 00 ENES 0 Mechanics of Materials Department of Civil and Environmental
More informationEE C245 ME C218 Introduction to MEMS Design Fall 2010
EE C245 ME C218 Introduction to MEMS Design Fall 2010 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture EE C245:
More informationLateral-torsional buckling of orthotropic rectangular section beams
ateral-torsional buckling of orthotropic rectangular section beams Summary Bambang SUYOAMONO Professor Department of Civil ngineering, Parahyangan Catholic University Bandung, Indonesia Adhijoso JONDO
More informationThe stiffness of plates
The stiffness of plates 1. Introduction The word plate is a collective term for elements in which forces can be transferred in two directions. Floors, walls, bridge slabs and laminates are all plates.
More informationA HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS
A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A. Kroker, W. Becker TU Darmstadt, Department of Mechanical Engineering, Chair of Structural Mechanics Hochschulstr. 1, D-64289 Darmstadt, Germany kroker@mechanik.tu-darmstadt.de,
More informationA refined shear deformation theory for bending analysis of isotropic and orthotropic plates under various loading conditions
JOURNAL OF MATERIALS AND ENGINRING STRUCTURES (015) 3 15 3 Research Paper A refined shear deformation theor for bending analsis of isotropic and orthotropic plates under various loading conditions Bharti
More informationDISTORTION ANALYSIS OF TILL -WALLED BOX GIRDERS
Nigerian Journal of Technology, Vol. 25, No. 2, September 2006 Osadebe and Mbajiogu 36 DISTORTION ANALYSIS OF TILL -WALLED BOX GIRDERS N. N. OSADEBE, M. Sc., Ph. D., MNSE Department of Civil Engineering
More informationSimulation of Geometrical Cross-Section for Practical Purposes
Simulation of Geometrical Cross-Section for Practical Purposes Bhasker R.S. 1, Prasad R. K. 2, Kumar V. 3, Prasad P. 4 123 Department of Mechanical Engineering, R.D. Engineering College, Ghaziabad, UP,
More informationParametric study on the transverse and longitudinal moments of trough type folded plate roofs using ANSYS
American Journal of Engineering Research (AJER) e-issn : 2320-0847 p-issn : 2320-0936 Volume-4 pp-22-28 www.ajer.org Research Paper Open Access Parametric study on the transverse and longitudinal moments
More informationSlender Structures Load carrying principles
Slender Structures Load carrying principles Basic cases: Extension, Shear, Torsion, Cable Bending (Euler) v017-1 Hans Welleman 1 Content (preliminary schedule) Basic cases Extension, shear, torsion, cable
More informationAdvanced Vibrations. Distributed-Parameter Systems: Exact Solutions (Lecture 10) By: H. Ahmadian
Advanced Vibrations Distributed-Parameter Systems: Exact Solutions (Lecture 10) By: H. Ahmadian ahmadian@iust.ac.ir Distributed-Parameter Systems: Exact Solutions Relation between Discrete and Distributed
More informationDEPARTMENT OF CIVIL ENGINEERING
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING SUBJECT: CE 2252 STRENGTH OF MATERIALS UNIT: I ENERGY METHODS 1. Define: Strain Energy When an elastic body is under the action of external
More informationARTICLE IN PRESS. Physica E
Physica E 41 (2009) 1651 1655 Contents lists available at ScienceDirect Physica E journal homepage: www.elsevier.com/locate/physe A general nonlocal beam theory: Its application to nanobeam bending, buckling
More information